Ok, all is well until [tex]\bigtriangleup[/tex]. I know this represents the laplacian. What is the formulation of the Laplacian for this since it is a vector? Is it just simply the second partials dot product with the respective terms of the vector? Or is it something else?

edit: changed text where I say problem is [tex] \bigtriangledown[/tex] to the appropriate [tex]\bigtriangleup[/tex]

Well, that is an identity for the operators. But why don't you like the idea of the usual Laplacian acting on a vector? It's just a derivative operator, which is allowable on vectors as long as you remember that the basis vectors also have to be differentiated.